Yeah, it'll take a little while to get the right sized collection but you'll eventually get to a point where you can craft plenty of different deck types. Currently, I have 10 different decks for example.
Next week, when the nex expansion launches, there's 13 free packs coming for everyone, so that should help a fair bit.

No, you can only have one class. Only certain cards are class specific though, so most cards can be used in multiple decks. There are no neutral spells though, only minions (at least, there aren't at the moment.

You cannot have other classes' cards in your base deck. Many many cards will put random cards into your hand though, which is good for surprising opponents with an unexpected spell or minion.

Loads of cards are total crap, and you will get them from packs and such. You candisenchant them for crafting dust, and craft cards you actually do want with said dust - obviously, you get less dust from a card than it would take to create an equivalent rarity card.

The size of your deck is always consistent - you start off with 30 cards in any deck. During a game you may shuffle cards into your deck though, as well as the aforementioned putting cards into your hand (from places other than your deck).

@Dominoid: 13 free decks for everyone? I missed that nugget! Where from?

I have a few GGers battletags, gief the rest of yours pl0x.
(I'm JKFerret#2359 btw)